Exercice 1 : (8 points)

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Devoir commun de Terminale Mai 2021 Correction Lycée Français Jules Supervielle

1 Exercice 1 : (8 points)

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5 5. a. Étudier la convexité de 𝑓.

𝑓^′(𝑥) = (−1 2𝑥 +1

2) 𝑒⁻¹²^𝑥 𝑒𝑠𝑡 𝑑é𝑟𝑖𝑣𝑎𝑏𝑙𝑒 𝑑𝑢 𝑡𝑦𝑝𝑒 (𝑢𝑣)^′= 𝑢^′𝑣 + 𝑢𝑣^′ 𝑓^′′(𝑥) = −1

2× 𝑒⁻¹²^𝑥+ (−1 2𝑥 +1

2) × −1 2𝑒⁻¹²^𝑥

= −1

2× 𝑒⁻¹²^𝑥+ (1 4𝑥 −1

4) × 𝑒⁻¹²^𝑥= (−1 2+1

4𝑥 −1

4) × 𝑒⁻¹²^𝑥 = (1 4𝑥 −3

4) 𝑒⁻¹²^𝑥 𝑃𝑜𝑢𝑟 𝑡𝑜𝑢𝑡 𝑥 ∈ [0; +∞[, 𝑒⁻¹²^𝑥 > 0, 𝑑𝑜𝑛𝑐 𝑓^′′𝑒𝑠𝑡 𝑑𝑢 𝑠𝑖𝑔𝑛𝑒 𝑑𝑒 1

4𝑥 −3 4 𝑥 0 3 + ∞ 𝑠𝑖𝑔𝑛𝑒 𝑑𝑒 𝑓′′ − 0 +

𝒄𝒐𝒏𝒗𝒆𝒙𝒊𝒕é 𝒅𝒆 𝒇 𝒄𝒐𝒏𝒄𝒂𝒗𝒆 | 𝒄𝒐𝒏𝒗𝒆𝒙𝒆

b. 𝑒𝑛 𝑥 = 3, 𝑓^′′𝑠^′𝑎𝑛𝑛𝑢𝑙𝑒 𝑒𝑛 𝑐ℎ𝑎𝑛𝑔𝑒𝑎𝑛𝑡 𝑑𝑒 𝑠𝑖𝑔𝑛𝑒

𝑑𝑜𝑛𝑐 𝑙𝑎 𝑐𝑜𝑢𝑟𝑏𝑒 C_𝑓 𝑎𝑑𝑚𝑒𝑡 𝑢𝑛 𝑝𝑜𝑖𝑛𝑡 𝑑^′𝑖𝑛𝑓𝑙𝑒𝑥𝑖𝑜𝑛 𝑒𝑛 (3; 𝑓(3)) = (𝟑; 𝟒𝒆⁻^𝟑^𝟐) c. 𝐿𝑎 𝑇𝑎𝑛𝑔𝑒𝑛𝑡𝑒 à C_𝑓 𝑎𝑢 𝑝𝑜𝑖𝑛𝑡 𝑑^′𝑎𝑏𝑠𝑐𝑖𝑠𝑠𝑒 3 𝑒𝑠𝑡 𝑦 = 𝑓^′(3) × (𝑥 − 3) + 𝑓(3)

𝑑𝑜𝑛𝑐 𝑦 = −𝑒⁻³²(𝑥 − 3) + 4𝑒⁻³² 𝑒𝑡 𝑓𝑖𝑛𝑎𝑙𝑒𝑚𝑒𝑛𝑡 𝑦 = −𝑒⁻³²𝑥 + 7𝑒⁻³² 𝐿^′𝑎𝑥𝑒 𝑑𝑒𝑠 𝑎𝑏𝑠𝑐𝑖𝑠𝑠𝑒𝑠 𝑦 = 0 𝑒𝑠𝑡 𝑎𝑠𝑦𝑚𝑝𝑡ô𝑡𝑒 ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙𝑒 à 𝑙𝑎 𝑐𝑜𝑢𝑟𝑏𝑒 C_𝑓 𝑝𝑢𝑖𝑠𝑞𝑢𝑒 lim

𝑥→+∞𝑓(𝑥) = 0 𝐿𝑒 𝑝𝑜𝑖𝑛𝑡 𝑑^′𝑖𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑖𝑜𝑛 𝑑𝑒 𝑙𝑎 𝑡𝑎𝑛𝑔𝑒𝑛𝑡𝑒 𝑦 = −𝑒⁻³²𝑥 + 7𝑒⁻³² 𝑎𝑣𝑒𝑐 𝑙^′𝑎𝑥𝑒 𝑑𝑒𝑠 𝑎𝑏𝑠𝑐𝑖𝑠𝑠𝑒𝑠 𝑒𝑠𝑡 (𝑥; 0)

𝑎𝑣𝑒𝑐 𝑥 𝑞𝑢𝑖 𝑒𝑠𝑡 𝑙𝑎 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑑𝑒 𝑙^′é𝑞𝑢𝑎𝑡𝑖𝑜𝑛 − 𝑒⁻³²𝑥 + 7𝑒⁻³² = 0

−𝑒⁻³²𝑥 + 7𝑒⁻³² = 0 (−𝑥 + 7)𝑒⁻³²= 0

𝑒⁻³² > 0 𝑑𝑜𝑛𝑥 − 𝑥 + 7 = 0 𝑒𝑡 𝑓𝑖𝑛𝑎𝑙𝑒𝑚𝑒𝑛𝑡 𝑥 = 7 𝒑𝒐𝒊𝒏𝒕 𝒅^′𝒊𝒏𝒕𝒆𝒓𝒔𝒆𝒄𝒕𝒊𝒐𝒏 (𝟕; 𝟎)

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